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An order of magnitude is the class of scale or magnitude of any amount, where each class contains values of a fixed ratio to the class preceding it. In its most common usage, the amount being scaled is 10 and the scale is the (base 10) exponent being applied to this amount (therefore, to be an order of magnitude greater is to be 10 times as large). Such differences in order of magnitude can be measured on the logarithmic scale in "decades" (i.e. factors of ten). ^{[1]}
The order of magnitude of a physical quantity is its magnitude in powers of ten when that physical quantity is expressed in powers of ten with one digit to the left of decimal.
"We say two numbers have the same order of magnitude of a number if the big one divided by the little one is less than 10. For example, 23 and 82 have the same order of magnitude, but 23 and 820 do not."^{[2]}  John Baez
Contents

Orders of magnitude are generally used to make very approximate comparisons, and reflect very large differences. If two numbers differ by one order of magnitude, one is about ten times larger than the other. If they differ by two orders of magnitude, they differ by a factor of about 100. Two numbers of the same order of magnitude have roughly the same scale: the larger value is less than ten times the smaller value.
In words (long scale) 
In words (short scale) 
Prefix  Symbol  Decimal  Power of ten 
Order of magnitude 

quadrillionth  septillionth  yocto  y  0.000,000,000,000,000,000,000,001  10^{−24}  −24 
trilliardth  sextillionth  zepto  z  0.000,000,000,000,000,000,001  10^{−21}  −21 
trillionth  quintillionth  atto  a  0.000,000,000,000,000,001  10^{−18}  −18 
billiardth  quadrillionth  femto  f  0.000,000,000,000,001  10^{−15}  −15 
billionth  trillionth  pico  p  0.000,000,000,001  10^{−12}  −12 
milliardth  billionth  nano  n  0.000,000,001  10^{−9}  −9 
millionth  millionth  micro  µ  0.000,001  10^{−6}  −6 
thousandth  thousandth  milli  m  0.001  10^{−3}  −3 
hundredth  hundredth  centi  c  0.01  10^{−2}  −2 
tenth  tenth  deci  d  0.1  10^{−1}  −1 
one  one  –  –  1  10^{0}  0 
ten  ten  deca  da  10  10^{1}  1 
hundred  hundred  hecto  h  100  10^{2}  2 
thousand  thousand  kilo  k  1,000  10^{3}  3 
million  million  mega  M  1,000,000  10^{6}  6 
milliard  billion  giga  G  1,000,000,000  10^{9}  9 
billion  trillion  tera  T  1,000,000,000,000  10^{12}  12 
billiard  quadrillion  peta  P  1,000,000,000,000,000  10^{15}  15 
trillion  quintillion  exa  E  1,000,000,000,000,000,000  10^{18}  18 
trilliard  sextillion  zetta  Z  1,000,000,000,000,000,000,000  10^{21}  21 
quadrillion  septillion  yotta  Y  1,000,000,000,000,000,000,000,000  10^{24}  24 
The order of magnitude of a number is, intuitively speaking, the number of powers of 10 contained in the number. More precisely, the order of magnitude of a number can be defined in terms of the common logarithm, usually as the integer part of the logarithm, obtained by truncation. For example, the number 4,000,000 has a logarithm (in base 10) of 6.602; its order of magnitude is 6. When truncating, a number of this order of magnitude is between 10^{6} and 10^{7}. In a similar example, with the phrase "He had a sevenfigure income", the order of magnitude is the number of figures minus one, so it is very easily determined without a calculator to be 6. An order of magnitude is an approximate position on a logarithmic scale.
An orderofmagnitude estimate of a variable whose precise value is unknown is an estimate rounded to the nearest power of ten. For example, an orderofmagnitude estimate for a variable between about 3 billion and 30 billion (such as the human population of the Earth) is 10 billion. To round a number to its nearest order of magnitude, one rounds its logarithm to the nearest integer. Thus 4,000,000, which has a logarithm (in base 10) of 6.602, has 7 as its nearest order of magnitude, because "nearest" implies rounding rather than truncation. For a number written in scientific notation, this logarithmic rounding scale requires rounding up to the next power of ten when the multiplier is greater than the square root of ten (about 3.162). For example, the nearest order of magnitude for 1.7 × 10^{8} is 8, whereas the nearest order of magnitude for 3.7 × 10^{8} is 9. An orderofmagnitude estimate is sometimes also called a zeroth order approximation.
An orderofmagnitude difference between two values is a factor of 10. For example, the mass of the planet Saturn is 95 times that of Earth, so Saturn is two orders of magnitude more massive than Earth. Orderofmagnitude differences are called decades when measured on a logarithmic scale.
Other orders of magnitude may be calculated using bases other than 10. The ancient Greeks ranked the nighttime brightness of celestial bodies by 6 levels in which each level was the fifth root of one hundred (about 2.512) as bright as the nearest weaker level of brightness, so that the brightest level is 5 orders of magnitude brighter than the weakest, which can also be stated as a factor of 100 times brighter.
The different decimal numeral systems of the world use a larger base to better envision the size of the number, and have created names for the powers of this larger base. The table shows what number the order of magnitude aim at for base 10 and for base 1,000,000. It can be seen that the order of magnitude is included in the number name in this example, because bi means 2 and tri means 3 (these make sense in the long scale only), and the suffix illion tells that the base is 1,000,000. But the number names billion, trillion themselves (here with other meaning than in the first chapter) are not names of the orders of magnitudes, they are names of "magnitudes", that is the numbers 1,000,000,000,000 etc.
order of magnitude  is log_{10} of  is log_{1000000} of  short scale  long scale 

1  10  1,000,000  million  million 
2  100  1,000,000,000,000  trillion  billion 
3  1000  1,000,000,000,000,000,000  quintillion  trillion 
SI units in the table at right are used together with SI prefixes, which were devised with mainly base 1000 magnitudes in mind. The IEC standard prefixes with base 1024 were invented for use in electronic technology.
The ancient apparent magnitudes for the brightness of stars uses the base and is reversed. The modernized version has however turned into a logarithmic scale with noninteger values.
For extremely large numbers, a generalized order of magnitude can be based on their double logarithm or superlogarithm. Rounding these downward to an integer gives categories between very "round numbers", rounding them to the nearest integer and applying the inverse function gives the "nearest" round number.
The double logarithm yields the categories:
(the first two mentioned, and the extension to the left, may not be very useful, they merely demonstrate how the sequence mathematically continues to the left).
The superlogarithm yields the categories:
The "midpoints" which determine which round number is nearer are in the first case:
and, depending on the interpolation method, in the second case
For extremely small numbers (in the sense of close to zero) neither method is suitable directly, but of course the generalized order of magnitude of the reciprocal can be considered.
Similar to the logarithmic scale one can have a double logarithmic scale (example provided here) and superlogarithmic scale. The intervals above all have the same length on them, with the "midpoints" actually midway. More generally, a point midway between two points corresponds to the generalised fmean with f(x) the corresponding function log log x or slog x. In the case of log log x, this mean of two numbers (e.g. 2 and 16 giving 4) does not depend on the base of the logarithm, just like in the case of log x (geometric mean, 2 and 8 giving 4), but unlike in the case of log log log x (4 and 65536 giving 16 if the base is 2, but, otherwise).

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リンク元  「桁」「digit」 
関連記事  「order」「magnitude」 